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Factors, Multiples & Primes

Almost every number skill higher up the course, from cancelling fractions to adding them, rests on being able to break a number into its building blocks. This is where those blocks live.

MYP 4Standard MathsNumberCriteria A · C~9 min read

Written by Fayek Jawad Haider · MYP student

Factors and multiples

These two words get muddled all the time, so pin them down first. A factor of a number divides into it exactly, with no remainder. A multiple is what you get when you multiply that number by a whole number. Factors sit inside a number; multiples stretch out beyond it.

Factor
A whole number that divides into another exactly. The factors of 12 are 1, 2, 3, 4, 6 and 12.
Multiple
The result of multiplying a number by a whole number. The first four multiples of 12 are 12, 24, 36 and 48.

To list every factor, work in pairs from the outside in: \(1\times12\), \(2\times6\), \(3\times4\). Once the pairs start to meet in the middle, you have them all. Working in pairs is the trick that stops you missing one.

Factors are finite, multiples are not

A number has a fixed, countable list of factors, but its multiples go on for ever. If a question asks you to "list all", it can only mean factors.

Prime numbers

A prime number has exactly two factors: 1 and itself. Nothing else divides into it. The primes start 2, 3, 5, 7, 11, 13, 17, 19, and carry on for ever.

Two catches worth remembering. First, 1 is not prime, because it has only one factor (itself), not two. Second, 2 is the only even prime, since every other even number is divisible by 2 as well as 1 and itself.

Prime number
A whole number greater than 1 with exactly two factors, 1 and the number itself.

Numbers that are not prime (and are bigger than 1) are called composite: they can be built by multiplying smaller numbers together. That building idea is the whole point of the next section.

Prime factorisation

Every whole number bigger than 1 can be written as a product of primes, and there is only one way to do it (ignoring the order). This unique fingerprint is called its prime factorisation. A factor tree is the neatest way to find it: keep splitting into a factor pair until every branch ends on a prime.

Worked example

Write 60 as a product of its prime factors.

1
Split 60 into any factor pair, say \(60 = 6 \times 10\).
2
Neither 6 nor 10 is prime, so split again: \(6 = 2 \times 3\) and \(10 = 2 \times 5\).
3
Every branch now ends on a prime: 2, 3, 2, 5. Collect them: \(60 = 2 \times 2 \times 3 \times 5\).
\(60 = 2^2 \times 3 \times 5\)

Any starting pair gives the same answer. Had you begun with \(60 = 4 \times 15\), you would still land on \(2^2 \times 3 \times 5\). Writing repeated primes as a power, \(2\times2 = 2^2\), keeps the final line tidy.

HCF and LCM

Once you have the prime factorisations, two useful numbers drop out almost for free.

Highest Common Factor (HCF)
The largest number that divides into both. Take each prime the two numbers share, using the lower power, and multiply.
Lowest Common Multiple (LCM)
The smallest number both divide into. Take every prime that appears in either number, using the higher power, and multiply.
Worked example

Find the HCF and LCM of 24 and 36.

1
Factorise each: \(24 = 2^3 \times 3\) and \(36 = 2^2 \times 3^2\).
2
HCF: take the shared primes at their lower power. Both share a 2 (lower power \(2^2\)) and a 3 (lower power \(3^1\)): \(2^2 \times 3 = 12\).
3
LCM: take every prime at its higher power. Highest power of 2 is \(2^3\), highest power of 3 is \(3^2\): \(2^3 \times 3^2 = 8 \times 9 = 72\).
\(\text{HCF} = 12\), \(\quad \text{LCM} = 72\)

Where this is assessed

This topic lives in Criterion A (knowing and applying the method) and Criterion C (setting out the factor tree and answer clearly). A quick sense check: the HCF can never be bigger than the smaller number, and the LCM can never be smaller than the larger number.

Check yourself

1. List all the factors of 18. +

Work in pairs: \(1\times18\), \(2\times9\), \(3\times6\). The factors are 1, 2, 3, 6, 9 and 18.

2. Write 84 as a product of its prime factors. +

Split it down: \(84 = 4 \times 21 = (2\times2)\times(3\times7)\). Collecting the primes gives \(84 = \mathbf{2^2 \times 3 \times 7}\).

3. Find the HCF and LCM of 12 and 18. +

Factorise: \(12 = 2^2\times3\) and \(18 = 2\times3^2\). HCF takes shared primes at the lower power: \(2\times3 = \mathbf{6}\). LCM takes every prime at the higher power: \(2^2\times3^2 = \mathbf{36}\).

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